Study of a qualitative model for combustion waves: flames, detonations, and deflagration-to-detonation transition. (English) Zbl 07833800
Summary: We analyze a simplified one-dimensional model of combustion that includes the effects of compressibility, diffusion, heat conduction, viscosity, and exothermic heat release. Using numerical simulations, we show that the model predicts both detonation-like and deflagration-like traveling waves. Importantly, it also predicts spontaneous transition from deflagration to detonation in the problem of hot-spot ignition from a closed end of a channel.
MSC:
| 76-XX | Fluid mechanics |
Keywords:
detonation; deflagration to detonation transition; detonation analog; reactive Navier-Stokes equationsReferences:
| [1] | Fickett, W., Detonation in miniature, Amer J Phys, 47, 12, 1050-1059, 1979 |
| [2] | Majda, A., A qualitative model for dynamic combustion, SIAM J Appl Math, 41, 1, 70-93, 1981 · Zbl 0472.76075 |
| [3] | Radulescu, M. I.; Tang, J., Nonlinear dynamics of self-sustained supersonic reaction waves: Fickett’s detonation analogue, Phys Rev Lett, 107, 16, Article 164503 pp., 2011 |
| [4] | Kasimov, A. R.; Faria, L. M.; Rosales, R. R., Model for shock wave chaos, Phys Rev Lett, 110, 10, Article 104104 pp., 2013 |
| [5] | Fickett, W.; Davis, W. C., Detonation: theory and experiment, 2000, Dover Publications |
| [6] | Williams, F. A., Combustion theory, 2018, CRC Press |
| [7] | Shepherd, J. E.; Lee, J. H.S., On the transition from deflagration to detonation, (Major research topics in combustion, 1992, Springer: Springer New York), 439-487 |
| [8] | Khokhlov, A.; Oran, E.; Thomas, G., Numerical simulation of deflagration-to-detonation transition: the role of shock-flame interactions in turbulent flames, Combust Flame, 117, 1-2, 323-339, 1999 |
| [9] | Smirnov, N. N.; Tyurnikov, M. V., Experimental investigation of deflagration to detonation transition in hydrocarbon-air gaseous mixtures, Combust Flame, 100, 4, 661-668, 1995 |
| [10] | Poludnenko, A. Y.; Gardiner, T. A.; Oran, E. S., Spontaneous transition of turbulent flames to detonations in unconfined media, Phys Rev Lett, 107, 5, Article 054501 pp., 2011 |
| [11] | Poludnenko, A. Y.; Chambers, J.; Ahmed, K.; Gamezo, V. N.; Taylor, B. D., A unified mechanism for unconfined deflagration-to-detonation transition in terrestrial chemical systems and type Ia supernovae, Science, 366, 6465, eaau7365, 2019 |
| [12] | Liberman, M.; Ivanov, M.; Kiverin, A.; Kuznetsov, M.; Chukalovsky, A.; Rakhimova, T., Deflagration-to-detonation transition in highly reactive combustible mixtures, Acta Astronaut, 67, 7-8, 688-701, 2010 |
| [13] | Kuznetsov, M.; Liberman, M.; Matsukov, I., Experimental study of the preheat zone formation and deflagration to detonation transition, Combust Sci Technol, 182, 11-12, 1628-1644, 2010 |
| [14] | Zel’dovich, Y.; Librovich, V.; Makhviladze, G.; Sivashinsky, G., On the development of detonation in a non-uniformly preheated gas, Acta Astronaut, 15, 5-6, 313-321, 1970 |
| [15] | Oran, E. S.; Gamezo, V. N., Origins of the deflagration-to-detonation transition in gas-phase combustion, Combust Flame, 148, 1-2, 4-47, 2007 |
| [16] | Krivosheyev, P.; Novitski, A.; Penyazkov, O., Flame front dynamics, shape and structure on acceleration and deflagration-to-detonation transition, Acta Astronaut, 204, 692-704, 2022 |
| [17] | Deshaies, B.; Joulin, G., Flame-speed sensitivity to temperature changes and the deflagration-to-detonation transition, Combust Flame, 77, 2, 201-212, 1989 |
| [18] | Brailovsky, I.; Sivashinski, G. I., On deflagration-to-detonation transition, Combust Sci Technol, 130, 1-6, 201-231, 1997 |
| [19] | Kagan, L.; Gordon, P.; Sivashinsky, G., An asymptotic study of the transition from slow to fast burning in narrow channels, Proc Combust Inst, 35, 1, 913-920, 2015 |
| [20] | Kurdyumov, V. N.; Matalon, M., Self-accelerating flames in long narrow open channels, Proc Combust Inst, 35, 1, 921-928, 2015 |
| [21] | Clavin, P.; Tofaili, H., A one-dimensional model for deflagration to detonation transition on the tip of elongated flames in tubes, Combust Flame, 232, Article 111522 pp., 2021 |
| [22] | Clavin, P., Mechanism of deflagration-to-detonation transition in gas, 2023, arXiv preprint arXiv:2307.05485 |
| [23] | Clavin, P., Intrinsic transition mechanism to detonation of gaseous laminar flames in tubes, Comptes Rendus. Mécanique, 351, G2, 401-427, 2023 |
| [24] | Faria, L. M.; Kasimov, A. R.; Rosales, R. R., Theory of weakly nonlinear self-sustained detonations, J Fluid Mech, 784, 163-198, 2015 · Zbl 1382.76155 |
| [25] | Kabanov, D. I.; Kasimov, A. R., A minimal hyperbolic system for unstable shock waves, Commun Nonlinear Sci Numer Simul, 70, 282-301, 2019 · Zbl 1464.35157 |
| [26] | Bourlioux, A.; Majda, A. J.; Roytburd, V., Theoretical and numerical structure for unstable one-dimensional detonations, SIAM J Appl Math, 51, 2, 303-343, 1991 · Zbl 0731.76076 |
| [27] | Yao, J.; Stewart, D. S., On the dynamics of multi-dimensional detonation, J Fluid Mech, 309, 225-275, 1996 · Zbl 0887.76096 |
| [28] | Clavin, P.; Williams, F. A., Dynamics of planar gaseous detonations near Chapman-Jouguet conditions for small heat release, Combust Theory Model, 6, 1, 127-139, 2002 · Zbl 0999.80011 |
| [29] | Kasimov, A. R.; Stewart, D. S., Asymptotic theory of evolution and failure of self-sustained detonations, J Fluid Mech, 525, 161-192, 2005 · Zbl 1065.76129 |
| [30] | Bdzil, J. B.; Stewart, D. S., Theory of detonation shock dynamics, (Shock waves science and technology library, vol. 6: detonation dynamics, 2012, Springer: Springer Berlin Heidelberg), 373-453 |
| [31] | Faria, L. M.; Lau-Chapdelaine, S.; Kasimov, A. R.; Rosales, R. R., A toy model for detonations and flames, (26th international colloquium on the dynamics of explosions and reactive systems, 2017), 6 |
| [32] | Kapila, A. K.; Matkowsky, B. J.; Van Harten, A., An asymptotic theory of deflagrations and detonations I. The steady solutions, SIAM J Appl Math, 43, 3, 491-519, 1983 · Zbl 0512.76116 |
| [33] | Matkowsky, B.; Sivashinsky, G., An asymptotic derivation of two models in flame theory associated with the constant density approximation, SIAM J Appl Math, 37, 3, 686-699, 1979 · Zbl 0437.76065 |
| [34] | Weber, R.; Mercer, G.; Sidhu, H.; Gray, B., Combustion waves for gases \(( \operatorname{Le} = 1)\) and solids \(( \operatorname{Le} \to \infty )\), Proc R Soc Lond Ser A Math Phys Eng Sci, 453, 1960, 1105-1118, 1997 · Zbl 0885.76105 |
| [35] | Rosales, R. R.; Majda, A., Weakly nonlinear detonation waves, SIAM J Appl Math, 43, 5, 1086-1118, 1983 · Zbl 0572.76062 |
| [36] | Faria, L. M.; Rosales, R. R., Equation level matching: An extension of the method of matched asymptotic expansion for problems of wave propagation, Stud Appl Math, 139, 2, 265-287, 2017 · Zbl 1373.35243 |
| [37] | Krivosheyev, P.; Novitski, A.; Penyazkov, O., Evolution of the reaction front shape and structure on flame acceleration and deflagration-to-detonation transition, Russ J Phys Chem B, 16, 4, 661-669, 2022 |
| [38] | Kurdyumov, V. N.; Matalon, M., Flame acceleration in long narrow open channels, Proc Combust Inst, 34, 1, 865-872, 2013 |
| [39] | Kurdyumov, V. N.; Matalon, M., Effects of gas compressibility on the dynamics of premixed flames in long narrow adiabatic channels, Combust Theory Model, 20, 6, 1046-1067, 2016 · Zbl 1519.76049 |
| [40] | Bykov, V.; Koksharov, A.; Kuznetsov, M.; Zhukov, V., Hydrogen-oxygen flame acceleration in narrow open ended channels, Combust Flame, 238, Article 111913 pp., 2022 |
| [41] | Yanenko, N. N., The method of fractional steps, 1971, Springer: Springer Berlin Heidelberg · Zbl 0209.47103 |
| [42] | Colella, P.; Majda, A.; Roytburd, V., Theoretical and numerical structure for reacting shock waves, SIAM J Sci Stat Comput, 7, 4, 1059-1080, 1986 · Zbl 0633.76060 |
| [43] | Oran, E. S.; Boris, J. P., Numerical simulation of reactive flow, 2001, Cambridge University Press · Zbl 0980.76002 |
| [44] | Leveque, R. J., Finite volume methods for hyperbolic problems, 2004, Cambridge University Press |
| [45] | Strang, G., On the construction and comparison of difference schemes, SIAM J Numer Anal, 5, 3, 506-517, 1968 · Zbl 0184.38503 |
| [46] | Liu, X.-D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J Comput Phys, 115, 1, 200-212, 1994 · Zbl 0811.65076 |
| [47] | Shu, C.-W., High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM Rev, 51, 1, 82-126, 2009 · Zbl 1160.65330 |
| [48] | Gottlieb, S.; Shu, C.-W., Total variation diminishing Runge-Kutta schemes, Math Comp, 67, 221, 73-85, 1998 · Zbl 0897.65058 |
| [49] | Henrick, A.; Aslam, T.; Powers, J., Simulations of pulsating one-dimensional detonations with true fifth order accuracy, J Comput Phys, 213, 1, 311-329, 2006 · Zbl 1146.76669 |
| [50] | Goldin, A. Y.; Medvedeva, T. O.; Kasimov, A. R., Mode locking in gaseous detonation propagation in a channel with periodically varying friction, Phys Fluids, 34, 9, Article 096104 pp., 2022 |
| [51] | Goldin, A. Y.; Kasimov, A. R., Synchronization of detonations: Arnold tongues and devil’s staircases, J Fluid Mech, 946, R1, 2022 · Zbl 1500.76107 |
| [52] | Kasimov, A. R.; Goldin, A. Y., Resonance and mode locking in gaseous detonation propagation in a periodically nonuniform medium, Shock Waves, 31, 8, 841-849, 2021 |
| [53] | Bezanson, J.; Karpinski, S.; Shah, V. B.; Edelman, A., Julia: A fast dynamic language for technical computing, 2012, arXiv:1209.5145 |
| [54] | Bezanson, J.; Edelman, A.; Karpinski, S.; Shah, V. B., Julia: A fresh approach to numerical computing, SIAM Rev, 59, 1, 65-98, 2017 · Zbl 1356.68030 |
| [55] | Kurdyumov, V. N.; Matalon, M., Critical conditions for flame acceleration in long adiabatic channels closed at their ignition end, Proc Combust Inst, 36, 1, 1549-1557, 2017 |
| [56] | Kurdyumov, V. N.; Matalon, M., Influence of heat-loss on compressibility-driven flames propagating from the closed end of a long narrow duct, Combust Flame, 214, 1-13, 2020 |
| [57] | Kagan, L.; Sivashinsky, G., Parametric transition from deflagration to detonation: Runaway of fast flames, Proc Combust Inst, 36, 2, 2709-2715, 2017 |
| [58] | Melguizo-Gavilanes, J.; Ballossier, Y.; Faria, L. M., Experimental and theoretical observations on DDT in smooth narrow channels, Proc Combust Inst, 38, 3, 3497-3503, 2021 |
| [59] | Arndt, D.; Bangerth, W.; Feder, M.; Fehling, M.; Gassmöller, R.; Heister, T., The Library, version 9.4, J Numer Math, 30, 3, 231-246, 2022 · Zbl 07590497 |
| [60] | Hindmarsh, A. C.; Brown, P. N.; Grant, K. E.; Lee, S. L.; Serban, R.; Shumaker, D. E., SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers, ACM Trans Math Software, 31, 3, 363-396, 2005 · Zbl 1136.65329 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
